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Understanding the global dynamics of a robot controller, such as identifyingattractors and their regions of attraction (RoA), is important for safedeployment and synthesizing more effective hybrid controllers. This paperproposes a topological framework to analyze the global dynamics of robotcontrollers, even data-driven ones, in an effective and explainable way. Itbuilds a combinatorial representation representing the underlying system'sstate space and non-linear dynamics, which is summarized in a directed acyclicgraph, the Morse graph. The approach only probes the dynamics locally byforward propagating short trajectories over a state-space discretization, whichneeds to be a Lipschitz-continuous function. The framework is evaluated giveneither numerical or data-driven controllers for classical robotic benchmarks.It is compared against established analytical and recent machine learningalternatives for estimating the RoAs of such controllers. It is shown tooutperform them in accuracy and efficiency. It also provides deeper insights asit describes the global dynamics up to the discretization's resolution. Thisallows to use the Morse graph to identify how to synthesize controllers to formimproved hybrid solutions or how to identify the physical limitations of arobotic system.

Morse Graphs: Topological Tools for Analyzing the Global Dynamics of Robot Controllers